Note: Steps below are executed on Mingw and should be used with little or no modifications on Linux.

Go and download the archive that contains the modified Three files from here. put them into a directory say in your home folder with the following path: ~/objc/json

cd ~/objc/json and issue the following command to build the files:
gcc -c `gnustep-config --objc-flags` *.m
If no errors reported, this means the build process succeed. and the object file has been generated.

Now let's put this files inside the gnustep installation, to do so, mv the the .h files NSJSONSerialization.h and GSFastEnumeration.h to /GNUstep/System/Library/Headers/Foundation

Now let's create a static library and put it inside gnustep libraries path:
Use the following command:
ar q libNSJSONSerialization.a NSJSONSerialization.o
Then move the generated libNSJSONSerialization.a file to /GNUstep/System/Library/Libraries/

Note that, we now copied two.h files and one.a file into gnustep installation directories. You now safe to remove all the directory ~/objc/json and all of its contents.

Now You are done installing the NSJSONSerialization as a static lib into gnustep.

Now let's write a test program. Now create a brand new directory into your home directory say ~/objc/json_test and create a file named json_main.m in this directory.
Put the following code in it:

Now use the following command to compile it:
gcc `gnustep-config --objc-flags` *.m -L /GNUstep/System/Library/Libraries -lNSJSONSerialization -lobjc -lgnustep-base
[The inclusion for the libraries should be exactly in this order]